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Hamiltonian paths in Cartesian powers of directed cycles. (English) Zbl 1032.05069
Summary: The vertex set of the $$k$$th Cartesian power of a directed cycle of length $$m$$ can be naturally identified with the abelian group $$(\mathbb Z_m)^k$$. For any two elements $$u=(u_1,\dots,u_k)$$ and $$v=(v_1,\dots,v_k)$$ of $$(\mathbb Z_m)^k$$, it is easy to see that if there is a Hamiltonian path from $$u$$ to $$v$$, then $$u_1+\cdots+u_k \equiv v_1+\cdots+v_k+1\pmod m$$. We prove the convers, unless $$k=2$$ and $$m$$ is odd.

##### MSC:
 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs
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